Timeline for Point in Polygon algorithm from the viewpoint of a robot

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Dec 24, 2010 at 21:09	comment	added	Mark Bennet		@sleepless-in-beantown, agreed the interior path is square, and $2 \pi$ is not correct. There are some subtleties in definition too. For example the inner path in the square is uniformly distance 1 from the outer path, in the sense that the minimum distance from any point on the inner path to the nearest point on the square is 1. But the original square is not uniformly distance 1 from the inner path (look at the corners). [there is probably a better word than 'uniformly']. I suspect that the $2 \pi$ figure applies when there is uniform distance 1 both ways in the above sense.
Dec 24, 2010 at 20:26	history	edited	sleepless in beantown	CC BY-SA 2.5	fixed formula of outer level set length
Dec 24, 2010 at 20:24	comment	added	sleepless in beantown		@Mark-Bennet, you are correct that the exterior level set is $2\pi$ larger. My mistake. The interior level set, however, is a smaller square of edge length $2r-2$ for $r \gt 0$.
Dec 24, 2010 at 19:44	comment	added	Mark Bennet		On the square example, outside path, why is your path not composed of straight lines parallel to the sides, plus a quarter-circle at each corner? This would give $2 \pi$.
Dec 24, 2010 at 19:04	history	answered	sleepless in beantown	CC BY-SA 2.5